Question

consider the following proof of the base angles theorem. which statement should fill in the blank? proof: given isosceles △abc with $overline{ab}congoverline{bc}$, i can construct $overleftrightarrow{bd}$, the angle bisector of ∠b. _____________________. i also know that line segments are congruent to themselves, so $overline{bd}congoverline{bd}$ by the reflexive property of congruence. i now have two pairs of sides and an included angle that are congruent, so i know that △abd≅△cbd by the sas congruence theorem. finally, corresponding parts of congruent triangles are congruent by the cpctc theorem, so ∠a≅∠c. (1 point) then, by the definition of an isosceles triangle, i know that $overline{ab}congoverline{ca}$. then, by the definition of an angle bisector, i know that ∠abd≅∠cbd. then, by the definition of a mid - point, i know that $overline{ad}congoverline{dc}$. then, by the definition of an angle bisector i know that ∠bac≅∠bca.

Explanation:

Step1: Recall angle - bisector property

We are constructing the angle - bisector of $\angle B$. By the definition of an angle bisector, it divides an angle into two congruent angles. So when $\overleftrightarrow{BD}$ is the angle bisector of $\angle B$, we have $\angle ABD\cong\angle CBD$.

Step2: Analyze the proof structure

In the proof of $\triangle ABD\cong\triangle CBD$ using the SAS (Side - Angle - Side) Congruence Theorem, we already know that $AB\cong BC$ and $BD\cong BD$. The missing piece for the "angle" part of SAS is the congruence of the angles formed by the bisector.

Answer:

Then, by the definition of an angle bisector, I know that $\angle ABD\cong\angle CBD$